No abstract

No abstract

No abstract

The categorical theory of closure operators is used to introduce and study separated, complete and compact objects with respect to the Zariski closure operator naturally defined in any category X (A, Ω) obtained by a given complete category X (endowed with a proper factorization structure for morphisms) and by a given X -algebra (A, Ω) by forming the affine X -objects modelled by (A, Ω).Several basic examples are provided.

We introduce and study a concept of neighborhoods with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject lattices, naturally generalizes the classical neighborhoods in topological spaces and we show that it behaves accordingly. We investigate also separation and compactness dened in a natural way by the help of the neighborhoods introduced.